Hermitian-Singer Functional and Differential Codes
Gábor Korchmáros, Federico Romaniello, Valentino Smaldore
TL;DR
The paper constructs Hermitian Singer algebraic-geometric codes by taking the orbit of a Singer subgroup in $PGU(3,q)$ to form a $|G|=q^2-q+1$ divisor on the Hermitian curve ${\mathcal H}(q)$, and analyzes both the functional code $C_{\mathcal L}(\mathtt D,\mathtt G)$ and its dual, the differential code $C_{\Omega}(\mathtt D,\mathtt G)$. By leveraging the action of the Singer subgroup and the geometry of a related subplane $\Pi$ containing a family of Hermitian curves $H_t$, the authors derive explicit dimensions, designed distances, and sometimes equality with the actual minimum distance, including a subcode $\Lambda$ with notable parameters and a generalized $\lambda\mathtt G$ family. They provide concrete parameter sets: for $G$ of size $q^2-q+1$ and $D$ of size $q^3+1-|G|$, the Hermitian Singer functional code has length $n=q^3-q^2+q$, dimension $k=\frac{1}{2}(q^2-q)+2$, and designed distance $\delta=q^3-2q^2+2q-1$, with a subcode of dimension $\frac{1}{2}(q^2-q)$ attaining a close (sometimes exact) minimum distance; the dual differential code has designed distance $3$ and a proven bound $d\le\frac{1}{2}(q^2-q)+2$. Computational results for $q=3$ and $q=4$ via a projective-subplane realization and MAGMA confirm the theoretical bounds and show when the upper bounds are strict, illustrating the practicality of these constructions for quasi-cyclic Hermitian codes.
Abstract
Algebraic geometry codes on the Hermitian curve have been the subject of several papers, since they happen to have good performances and large automorphism groups. Here, those arising from the Singer cycle of the Hermitian curve are investigated.